Exercises

1. For each of the following series,
   • decide whether the series converges or diverges, and state how you know;
   • if the series converges, state whether you know its value, and give that value if you know one; and
   • if the series converges but you don't know its value, estimate the value with your computer or calculator.

   a.   `sum_(n=1)^oo (text[(]-1text[)]^(n+1))/n^2`	b.   `sum_(n=1)^oo (text[(]-1text[)]^n)/sqrt(n)`	c.   `sum_(n=1)^oo (text[(]-1text[)]^(n+1))/2^n`
   d.   `sum_(n=1)^oo (text[(]-1text[)]^n)/(sqrt(2))^n`	e.   `sum_(n=1)^oo 1/(n+1)`	f.   `sum_(n=1)^oo (text[(]-1text[)]^n n)/(n+10)`
   g.   `sum_(n=1)^oo (text[(]-1text[)]^(n+1))/(sin n)`	h.   `sum_(n=1)^oo (text[(]-1text[)]^n)/(2n+3)`

2. Here again is the Taylor series for the cosine function:

   `cos x=1-1/2 x^2+1/24 x^4-cdots+(text[(]-1text[)]^k)/(text[(]2ktext[)]!) x^(2k)+cdots`

   1. Explain why, for any fixed `x`, `lim_(k rarr oo) x^(2k)/(text[(]2ktext[)]!)=0`.
   2. Explain why, for any fixed `x` and for sufficiently large values of `k,`

      `x^(2k)/(text[(]2ktext[)]!)>x^(2k+2)/(text[(]2k+2text[)]!) `.

   3. Explain why the cosine series converges for every real number `x.`
3. 1. Estimate the size of the `n`th tail of the cosine series, i.e., the error after summing the terms numbered `0` to `n-1`.
   2. Calculate your error estimate for `x=5` and `n=7`.
   3. Calculate the sum of the first `7` terms of the series for `cos x` with `x=5`.
   4. Compare the sum in part (c) with `cos 5`. How close is the estimate in part (b) to the actual error?
4. Use the Ratio Test to determine the interval of convergence of each of the following series — even if you already know the interval. If the interval has endpoints, use some other test(s) to determine convergence or divergence at the endpoints.
   1. The exponential series
   2. The sine series
   3. The cosine series
   4. The logarithmic series
   5. The binomial series (See Exercise 10 in Section 10.4.)
5. Find the interval of convergence of each of the following series. If the interval has endpoints, determine convergence or divergence at the endpoints as best you can.

   a.   `sum_(k=1)^oo k^2 x^2`	b.   `sum_(k=0)^oo x^k/3^k`	c.   `sum_(k=1)^oo (k x^k)/2^k`
   d.   `sum_(k=0)^oo text[(]-1text[)]^k x^(2k+1)`	e.   `sum_(k=0)^oo (text[(]-1text[)]^k x^(2k))/(3^k(k+1))`	f.   `sum_(k=1)^oo (text[(]-1text[)]^(k+1) k! x^k)/2^k`
   g.   `sum_(k=0)^oo (text[(]-1text[)]^k x^(2k))/(k!)`	h.   `sum_(k=0)^oo (text[(]-1text[)]^k x^(2k))/(3^k(k+1)!)`	i.   `sum_(k=1)^oo (text[(]-1text[)]^(k+1) 10^k x^k)/(k!)`

6. 1. Determine how close `-(0.75+0.75^2/2+0.75^3/3+cdots+0.75^50/50)` is to `ln 0.25.` Use your computer or calculator to check your answer. (You may have to think about how to add a sum of 50 terms.)
   2. Determine how close `0.75+0.75^2/2+0.75^3/3+cdots+0.75^50/50` is to `ln 4`. Use your computer or calculator to check your answer.
   3. What logarithm is approximated by

      `0.65+0.65^2/2+0.65^3/3+cdots+0.65^50/50`?

      Determine how close the sum is to that logarithm. Use your computer or calculator to check your answer.
7. 1. Estimate the size of the `n`th tail of the Taylor series for `ln text[(] 1+x text[)]` with `x=-1`/`3`.
   2. Substitute `x=-1`/`3` in the logarithmic series and simplify to find an explicit expression for the series whose `n`th tail you estimated in part (a).
   3. Find an `n` for which the estimate in part (a) is less than `0.0001`.
   4. Add up `n` terms of the series in part (b), where `n` is the number you identified in part (c).
   5. What logarithm should be approximated to within `0.0001` by your answer to part (d)? Use your computer or calculator to check that the answer in (d) is within `0.0001` of the appropriate logarithm.
   6. How big is the actual error in your answer to part (d)? How good was your estimate of the error in part (a)?
8. Construct an alternating series in the following way: Take the positive terms to be the terms of the harmonic series, `1, 1/2, 1/3, 1/4, ...`. Take the negative terms to be the negatives of terms from a geometric series with ratio `1//2`, that is, `1, 1/2, 1/2^2, 1/2^3, ...`. Thus the alternating series starts out

   `1-1+1/2-1/2+1/3-1/4+1/4-1/8+1/5-1/16+cdots`.

   Notice that the terms strictly alternate in sign. Notice also that the limiting value for the terms is `0`.
   1. What is the limiting behavior of the partial sums? Why?
   2. Why doesn't this example contradict the Alternating Series Test?
9. 1. Using Example 3 as a model, find a bound for the error when the exponential power series `G text[(] x text[)]` is approximated by the `n`th partial sum `P_(n-1) text[(] x text[)]` on the interval `-a<=x<=a`. Your answer should involve `n` and `a`. Check that your formula gives the right estimate if `n=10` and `a=2`.
   2. Estimate the error in the 49th degree approximation on the interval `[-10,10]`.
   3. Would `P_49 text[(] x text[)]` be an adequate way to program the exponential function for a computer or calculator if we assume that only values for `-10<=x<=10` are needed? Explain.
10. Suppose `x` is any real number, and let `M` be any fixed integer bigger than `x`. For any `k` larger than `M`, write

    x k k ! = x ⋅ x ⋅ x ⋅⋅⋅ x 1 ⋅ 2 ⋅ 3 ⋅⋅⋅ M ×     x     ⋅     x     ⋅     x     ⋅⋅⋅     x ( M + 1 ) ⋅ ( M + 2 ) ⋅ ( M + 3 ) ⋅⋅⋅ k .

    Observe that the first fraction on the right has `M` factors in numerator and denominator, and the second fraction has `k-M` factors in numerator and denominator. Use this factorization to explain the limiting behavior of `x^k//k!` as `k rarr oo`.
11. In Exercise 11 of Section 10.4 you calculated five terms of a power series for `f text[(] x text[)] =sqrt(1+x)` and found that the terms alternate in sign after the first two. Now you know how to estimate the error when a partial sum of this series is used to approximate the total sum.
    1. Find a bound for the error if you use five terms of this series to approximate `sqrt(1.2)`.
    2. How does the bound in part (a) compare with the actual error you found in Exercise 11 of Section 10.4?
    3. How many terms of this series would be required to calculate `sqrt(1.2)` to three-decimal-place accuracy? Use your computer or calculator to check your answer.
12. 1. Find the first five terms of the Taylor series for `g text[(] x text[)] =sqrt(1+x^3)`. (You may have done this already in Exercise 12 of Section 10.4.)
    2. Calculate `int_0^(1//2) sqrt(1+x^3) dx`, accurate to three decimal places. Explain how you know — without relying on the integral function of your computer or calculator — that the answer has that accuracy. Use your computer or calculator to check your accuracy.
13. 1. Why doesn't the cube-root function `x^(1//3)` have a Taylor series at the reference point `x=0`?
    2. Find a Taylor series for the cube-root function at the reference point `x=1`.
    3. Find the interval of convergence of the series in part (b). If the interval has endpoints, determine what you can about convergence at the endpoints.
14. Here's an idea for getting better estimates from an alternating series. In Table E1 we give terms and partial sums of the alternating harmonic series for moderate-sized values of `n`, along with errors in estimating the total sum, which happens to be `ln 2`. Notice that each added term overshoots the eventual answer, on the high side if the term is positive, on the low side if it is negative. Since the terms are roughly the same size, we could get much closer by adding only half the next term to a given partial sum. Fill in the last column of the table to see what we mean. (Click here for a printable copy of the table on which you can record your work.)
    
    

    Table E1   Estimating ln 2 by corrected partial sums

    `n`	`text[(]-1text[)]^(n+1)text[/]n`	partial sum	|ln 2-partial sum|	partial sum+(next term)/2	|error|
    5	0.2000	0.7833	0.0902	0.7000	0.00685
    6	-0.1667	0.6167	0.0764
    7	0.1429	0.7595	0.0664
    8	-0.1250	0.6345	0.0586
    9	0.1111	0.7456	0.0525
    10	-0.1000	0.6456	0.0475

15. Use the idea of the preceding exercise to estimate the sum of the Leibniz series,

    1 - 1 3 + 1 5 - 1 7 + ⋅⋅⋅ + ( - 1 ) k 1 2 k + 1 + ⋅⋅⋅ . .

    Record your results in Table E2. The exact sum of this series happens to be `pi//4`. Fill in the last column to see how close your estimates in the next-to-last column really are. (Click here for a printable copy of the table on which you can record your work.)
    
    

    Table E2  Estimating `pi//4` by corrected partial sums

    `n`	`(text[(]-1text[)]^n)/(2n+1)`	partial sum	`|pitext[/]4-text(partial sum)|`	`text(partial sum)+1/2text[(next term)]`	`|text(error)|`
    5	-0.0909	0.7440	0.0414
    6
    7
    8
    9
    10

16. Use the idea of the two preceding exercises to estimate

    1 - 1 4 + 1 9 - 1 16 + ⋅⋅⋅ + ( - 1 ) k + 1 k 2 + ⋅⋅⋅ .

    Record your results in Table E3. How many digits in your last estimate do you think match the exact answer? Why? (Click here for a printable copy of the table on which you can record your work.)
    
    

    Table E3  Estimating a total sum by corrected partial sums

    `n`	`(text[(]-1text[)]^(n+1))/n^2`	`text(partial sum)`	`text(partial sum)+1/2text[(next term)]`
    5	0.0400	0.8386
    6
    7
    8
    9
    10